# On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Some Remarks on Symmetry Issues

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Applications

**Lemma**

**1.**

- (I)
- in the case $\mathbb{K}=\mathbb{R}$, there is $a\in \mathbb{C}$ with $g\left(s\right)={e}^{as}$ for $s\in \mathbb{R}$;
- (II)
- in the case $\mathbb{K}=\mathbb{C}$, there are $b,c\in \mathbb{C}$ with $g\left(s\right)={e}^{cs+b\overline{s}}$ for $s\in \mathbb{C}$.

**Lemma**

**2.**

**Proof.**

**Proposition**

**1.**

- (a)
- in the case $\mathbb{K}=\mathbb{R}$, $f\left(s\right)={F}_{0}\left({e}^{ids}\right)$ for $s\in \mathbb{R}$;
- (b)
- in the case $\mathbb{K}=\mathbb{C}$, $f\left(s\right)={F}_{0}\left({e}^{i(ds+\overline{d}\phantom{\rule{0.277778em}{0ex}}\overline{s}\phantom{\rule{0.166667em}{0ex}})}\right)$ for $s\in \mathbb{C}$.

**Proof.**

**Proposition**

**2.**

- (i)
- in the case $\mathbb{K}=\mathbb{R}$, $f\left(s\right)={H}^{-1}\left(ds\right)$ for $s\in \mathbb{R}$;
- (ii)
- in the case $\mathbb{K}=\mathbb{C}$, $f\left(s\right)={H}^{-1}(ds+\overline{d}\phantom{\rule{0.277778em}{0ex}}\overline{s})$ for $s\in \mathbb{C}$.

**Proof.**

**Proposition**

**3.**

- (A)
- A continuous function $f:I\to S$ fulfills the functional equation$$f(t\circ s)=f\left(t\right)\star f\left(s\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}s,t\in I,$$
- (B)
- Every continuous function $f:S\to I$ fulfilling the functional equation$$f(u\star v)=f\left(u\right)\circ f\left(v\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}u,v\in S,$$is constant.

**Proof.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 4. Auxiliary Results

**Remark**

**4.**

**Remark**

**5.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Corollary**

**1.**

**Lemma**

**6.**

- (a)
- $v\prec {v}^{2}$;
- (b)
- if $t,s\in S$, $t\prec s\prec v$, then ${t}^{2}\prec {s}^{2}$.

**Proof.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Remark**

**6.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Proof.**

**Lemma**

**13.**

- (a)
- For every $s\in S\setminus \left\{1\right\}$, there is $x\in D$, $x<1$, with $d\left(x\right)\prec s$.
- (b)
- The set $U:=\left\{d\right(x):x\in D,\phantom{\rule{0.166667em}{0ex}}x<M\}$ is dense in S.

**Proof.**

**Corollary**

**2.**

**Lemma**

**14.**

**Proof.**

**Lemma**

**15.**

- (a)
- if $x+y\le M$, then $\overline{d}(x+y)=\overline{d}\left(x\right)\circ \overline{d}\left(y\right)$;
- (b)
- if $M<x+y$, then $\overline{d}(x+y-M)=\overline{d}\left(x\right)\circ \overline{d}\left(y\right)$.

**Proof.**

## 5. Proof of Theorem 2

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Bajger, M.; Brzdęk, J.; El-hady, E.-s.; Jabłońska, E.
On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues. *Symmetry* **2020**, *12*, 1974.
https://doi.org/10.3390/sym12121974

**AMA Style**

Bajger M, Brzdęk J, El-hady E-s, Jabłońska E.
On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues. *Symmetry*. 2020; 12(12):1974.
https://doi.org/10.3390/sym12121974

**Chicago/Turabian Style**

Bajger, Mariusz, Janusz Brzdęk, El-sayed El-hady, and Eliza Jabłońska.
2020. "On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues" *Symmetry* 12, no. 12: 1974.
https://doi.org/10.3390/sym12121974